# Riemann theta functions

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##### 1: 21.2 Definitions
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###### §21.2(i) RiemannThetaFunctions
βΊFor numerical purposes we use the scaled Riemann theta function $\hat{\theta}\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$, defined by (Deconinck et al. (2004)), …Many applications involve quotients of Riemann theta functions: the exponential factor then disappears. … βΊ
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##### 2: 21.8 Abelian Functions
###### §21.8 Abelian Functions
βΊFor every Abelian function, there is a positive integer $n$, such that the Abelian function can be expressed as a ratio of linear combinations of products with $n$ factors of Riemann theta functions with characteristics that share a common period lattice. …
##### 3: 21.10 Methods of Computation
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###### §21.10(ii) RiemannThetaFunctions Associated with a Riemann Surface
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• Deconinck and van Hoeij (2001). Here a plane algebraic curve representation of the Riemann surface is used.

• ##### 4: 21.9 Integrable Equations
###### §21.9 Integrable Equations
βΊTypical examples of such equations are the Korteweg–de Vries equation … βΊβΊ βΊ
##### 5: 21.3 Symmetry and Quasi-Periodicity
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###### §21.3(ii) RiemannThetaFunctions with Characteristics
βΊ …For Riemann theta functions with half-period characteristics, …
##### 6: 21.1 Special Notation
βΊUppercase boldface letters are $g\times g$ real or complex matrices. βΊThe main functions treated in this chapter are the Riemann theta functions $\theta\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$, and the Riemann theta functions with characteristics $\theta\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{\beta}}}% \left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$. βΊThe function $\Theta(\boldsymbol{{\phi}}|\mathbf{B})=\theta\left(\boldsymbol{{\phi}}/(2\pi i% )\middle|\mathbf{B}/(2\pi i)\right)$ is also commonly used; see, for example, Belokolos et al. (1994, §2.5), Dubrovin (1981), and Fay (1973, Chapter 1).
##### 7: Sidebar 21.SB2: A two-phase solution of the Kadomtsev–Petviashvili equation (21.9.3)
βΊSuch a solution is given in terms of a Riemann theta function with two phases. …
##### 8: 21.4 Graphics
###### §21.4 Graphics
βΊFigure 21.4.1 provides surfaces of the scaled Riemann theta function $\hat{\theta}\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$, with … βΊFor the scaled Riemann theta functions depicted in Figures 21.4.221.4.5βΊ βΊ
##### 9: 21.5 Modular Transformations
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###### §21.5(i) RiemannThetaFunctions
βΊEquation (21.5.4) is the modular transformation property for Riemann theta functions. βΊThe modular transformations form a group under the composition of such transformations, the modular group, which is generated by simpler transformations, for which $\xi(\boldsymbol{{\Gamma}})$ is determinate: … βΊ
###### §21.5(ii) RiemannThetaFunctions with Characteristics
βΊFor explicit results in the case $g=1$, see §20.7(viii).
##### 10: 21.6 Products
###### §21.6 Products
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21.6.3 $\prod_{j=1}^{h}\theta\left(\sum_{k=1}^{h}T_{jk}\mathbf{z}_{k}\middle|% \boldsymbol{{\Omega}}\right)=\frac{1}{\mathcal{D}^{g}}\sum_{\mathbf{A}\in% \mathcal{K}}\sum_{\mathbf{B}\in\mathcal{K}}e^{2\pi i\operatorname{tr}\left[% \frac{1}{2}\mathbf{A}^{\mathrm{T}}\boldsymbol{{\Omega}}\mathbf{A}+\mathbf{A}^{% \mathrm{T}}[\mathbf{Z}+\mathbf{B}]\right]}\*\prod_{j=1}^{h}\theta\left(\mathbf% {z}_{j}+\boldsymbol{{\Omega}}\mathbf{a}_{j}+\mathbf{b}_{j}\middle|\boldsymbol{% {\Omega}}\right),$
βΊOn using theta functions with characteristics, it becomes … βΊ